Data assimilation algorithms combine information from observations and prior model information to obtain the most likely state of a dynamical system. The linearised weak-constraint four-dimensional variational assimilation problem can be reformulated as a saddle point problem, which admits more scope for preconditioners than the primal form. In this paper we design new terms which can be used within existing preconditioners, such as block diagonal and constraint-type preconditioners. Our novel preconditioning approaches: (i) incorporate model information whilst guaranteeing parallelism, and (ii) are designed to target correlated observation error covariance matrices. To our knowledge (i) has not previously been considered for data assimilation problems. We develop new theory demonstrating the effectiveness of the new preconditioners within Krylov subspace methods. Linear and non-linear numerical experiments reveal that our new approach leads to faster convergence than existing state-of-the-art preconditioners for a broader range of problems than indicated by the theory alone. We present a range of numerical experiments performed in serial, with further improvements expected if the highly parallelisable nature of the preconditioners is exploited.

翻译：数据同化算法通过融合观测信息与先验模型信息，获取动力系统最可能的状态。线性化弱约束四维变分同化问题可重新表述为鞍点问题，该形式相比原始形式具有更丰富的预条件子设计空间。本文针对现有预条件子（如块对角型与约束型预条件子）设计了可嵌入的新项。我们的创新预条件策略：（i）在保证并行性的同时融入模型信息，（ii）针对相关观测误差协方差矩阵进行优化设计。据我们所知，策略（i）此前尚未被应用于数据同化问题。我们发展了新理论，论证了此类预条件子在Krylov子空间方法中的有效性。线性与非线性数值实验表明，相比现有最优预条件子，新方法在更广泛的问题范围内实现了更快的收敛速度，适用范围超越理论分析的边界。我们呈现了一系列串行数值实验结果，而若充分利用预条件子的高度并行特性，预期可进一步提升性能。